## Abstract

Let script G sign → B be a bundle of compact Lie groups acting on a fiber bundle Y → B. In this paper we introduce and study gauge-equivariant K-theory groups K _{script G sign} ^{i}(Y). These groups satisfy the usual properties of the equivariant K-theory groups, but also some new phenomena arise due to the topological non-triviality of the bundle script G sign → B. As an application, we define a gauge-equivariant index for a family of elliptic operators (P _{b}) _{b∈B} invariant with respect to the action of script G sign → B, which, in this approach, is an element of K _{script G sign} ^{0}(B). We then give another definition of the gauge-equivariant index as an element of K _{0}(C* (script G sign)), the K-theory group of the Banach algebra C*(script G sign). We prove that K _{0}(C*(script G sign)) ≃ K _{script G sign} ^{0}(scrit G sign) and that the two definitions of the gauge-equivariant index are equivalent. The algebra C*(script G sign) is the algebra of continuous sections of a certain field of C*-algebras with non-trivial Dixmier-Douady invariant. The gauge-equivariant K-theory groups are thus examples of twisted K-theory groups, which have recently turned out to be useful in the study of Ramond-Ramond fields.

Original language | English (US) |
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Pages (from-to) | 185-218 |

Number of pages | 34 |

Journal | Transactions of the American Mathematical Society |

Volume | 356 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2004 |

## All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics